Theorem lgsdinn0 15727

Description: Variation on lgsdi 15716 valid for all M , N but only for positive A. (The exact location of the failure of this law is for A = -u 1, M = 0, and some N in which case ( -u 1 /L 0 ) = 1 but ( -u 1 /L N ) = -u 1 when -u 1 is not a quadratic residue mod N.) (Contributed by Mario Carneiro, 28-Apr-2016.)

Assertion

Ref	Expression
lgsdinn0	|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )

Proof of Theorem lgsdinn0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6009	. . . . . . . . 9 |- ( x = N -> ( A /L x ) = ( A /L N ) )
2	1	oveq1d 6016	. . . . . . . 8 |- ( x = N -> ( ( A /L x ) x. ( A /L 0 ) ) = ( ( A /L N ) x. ( A /L 0 ) ) )
3	2	eqeq2d 2241	. . . . . . 7 |- ( x = N -> ( ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) <-> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) ) )
4		sq1 10855	. . . . . . . . . . . . . . . . 17 |- ( 1 ^ 2 ) = 1
5	4	eqeq2i 2240	. . . . . . . . . . . . . . . 16 |- ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> ( A ^ 2 ) = 1 )
6		nn0re 9378	. . . . . . . . . . . . . . . . . 18 |- ( A e. NN0 -> A e. RR )
7		nn0ge0 9394	. . . . . . . . . . . . . . . . . 18 |- ( A e. NN0 -> 0 <_ A )
8		1re 8145	. . . . . . . . . . . . . . . . . . 19 |- 1 e. RR
9		0le1 8628	. . . . . . . . . . . . . . . . . . 19 |- 0 <_ 1
10		sq11 10834	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( 1 e. RR /\ 0 <_ 1 ) ) -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
11	8, 9, 10	mpanr12 439	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. RR /\ 0 <_ A ) -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
12	6, 7, 11	syl2anc 411	. . . . . . . . . . . . . . . . 17 |- ( A e. NN0 -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
13	12	adantr 276	. . . . . . . . . . . . . . . 16 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
14	5, 13	bitr3id 194	. . . . . . . . . . . . . . 15 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( ( A ^ 2 ) = 1 <-> A = 1 ) )
15	14	biimpa 296	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> A = 1 )
16	15	oveq1d 6016	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L x ) = ( 1 /L x ) )
17		1lgs 15722	. . . . . . . . . . . . . 14 |- ( x e. ZZ -> ( 1 /L x ) = 1 )
18	17	ad2antlr 489	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( 1 /L x ) = 1 )
19	16, 18	eqtrd 2262	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L x ) = 1 )
20	19	oveq1d 6016	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( ( A /L x ) x. ( A /L 0 ) ) = ( 1 x. ( A /L 0 ) ) )
21		nn0z 9466	. . . . . . . . . . . . . . 15 |- ( A e. NN0 -> A e. ZZ )
22	21	ad2antrr 488	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> A e. ZZ )
23		0z 9457	. . . . . . . . . . . . . 14 |- 0 e. ZZ
24		lgscl 15693	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> ( A /L 0 ) e. ZZ )
25	22, 23, 24	sylancl 413	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L 0 ) e. ZZ )
26	25	zcnd 9570	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L 0 ) e. CC )
27	26	mulid2d 8165	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( 1 x. ( A /L 0 ) ) = ( A /L 0 ) )
28	20, 27	eqtr2d 2263	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
29		lgscl 15693	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ x e. ZZ ) -> ( A /L x ) e. ZZ )
30	21, 29	sylan 283	. . . . . . . . . . . . . 14 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L x ) e. ZZ )
31	30	zcnd 9570	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L x ) e. CC )
32	31	adantr 276	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( A /L x ) e. CC )
33	32	mul01d 8539	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( ( A /L x ) x. 0 ) = 0 )
34	21	adantr 276	. . . . . . . . . . . . . 14 |- ( ( A e. NN0 /\ x e. ZZ ) -> A e. ZZ )
35		lgs0 15692	. . . . . . . . . . . . . 14 |- ( A e. ZZ -> ( A /L 0 ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) )
36	34, 35	syl 14	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L 0 ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) )
37		ifnefalse 3613	. . . . . . . . . . . . 13 |- ( ( A ^ 2 ) =/= 1 -> if ( ( A ^ 2 ) = 1 , 1 , 0 ) = 0 )
38	36, 37	sylan9eq 2282	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( A /L 0 ) = 0 )
39	38	oveq2d 6017	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( ( A /L x ) x. ( A /L 0 ) ) = ( ( A /L x ) x. 0 ) )
40	33, 39, 38	3eqtr4rd 2273	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
41		zsqcl 10832	. . . . . . . . . . . . 13 |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
42	34, 41	syl 14	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A ^ 2 ) e. ZZ )
43		1z 9472	. . . . . . . . . . . 12 |- 1 e. ZZ
44		zdceq 9522	. . . . . . . . . . . 12 |- ( ( ( A ^ 2 ) e. ZZ /\ 1 e. ZZ ) -> DECID ( A ^ 2 ) = 1 )
45	42, 43, 44	sylancl 413	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ x e. ZZ ) -> DECID ( A ^ 2 ) = 1 )
46		dcne 2411	. . . . . . . . . . 11 |- (DECID ( A ^ 2 ) = 1 <-> ( ( A ^ 2 ) = 1 \/ ( A ^ 2 ) =/= 1 ) )
47	45, 46	sylib 122	. . . . . . . . . 10 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( ( A ^ 2 ) = 1 \/ ( A ^ 2 ) =/= 1 ) )
48	28, 40, 47	mpjaodan 803	. . . . . . . . 9 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
49	48	ralrimiva 2603	. . . . . . . 8 |- ( A e. NN0 -> A. x e. ZZ ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
50	49	3ad2ant1 1042	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> A. x e. ZZ ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
51		simp3 1023	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
52	3, 50, 51	rspcdva 2912	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) )
53	52	adantr 276	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) )
54	21	3ad2ant1 1042	. . . . . . . . 9 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> A e. ZZ )
55	54, 23, 24	sylancl 413	. . . . . . . 8 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) e. ZZ )
56	55	zcnd 9570	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) e. CC )
57	56	adantr 276	. . . . . 6 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L 0 ) e. CC )
58		lgscl 15693	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ )
59	54, 51, 58	syl2anc 411	. . . . . . . 8 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ )
60	59	zcnd 9570	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. CC )
61	60	adantr 276	. . . . . 6 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L N ) e. CC )
62	57, 61	mulcomd 8168	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( A /L 0 ) x. ( A /L N ) ) = ( ( A /L N ) x. ( A /L 0 ) ) )
63	53, 62	eqtr4d 2265	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L 0 ) = ( ( A /L 0 ) x. ( A /L N ) ) )
64		oveq1 6008	. . . . . 6 |- ( M = 0 -> ( M x. N ) = ( 0 x. N ) )
65	51	zcnd 9570	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> N e. CC )
66	65	mul02d 8538	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 x. N ) = 0 )
67	64, 66	sylan9eqr 2284	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M x. N ) = 0 )
68	67	oveq2d 6017	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L ( M x. N ) ) = ( A /L 0 ) )
69		simpr 110	. . . . . 6 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> M = 0 )
70	69	oveq2d 6017	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L M ) = ( A /L 0 ) )
71	70	oveq1d 6016	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( A /L M ) x. ( A /L N ) ) = ( ( A /L 0 ) x. ( A /L N ) ) )
72	63, 68, 71	3eqtr4d 2272	. . 3 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
73		oveq2 6009	. . . . . . . 8 |- ( x = M -> ( A /L x ) = ( A /L M ) )
74	73	oveq1d 6016	. . . . . . 7 |- ( x = M -> ( ( A /L x ) x. ( A /L 0 ) ) = ( ( A /L M ) x. ( A /L 0 ) ) )
75	74	eqeq2d 2241	. . . . . 6 |- ( x = M -> ( ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) <-> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) ) )
76		simp2 1022	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
77	75, 50, 76	rspcdva 2912	. . . . 5 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) )
78	77	adantr 276	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) )
79		oveq2 6009	. . . . . 6 |- ( N = 0 -> ( M x. N ) = ( M x. 0 ) )
80	76	zcnd 9570	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> M e. CC )
81	80	mul01d 8539	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( M x. 0 ) = 0 )
82	79, 81	sylan9eqr 2284	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M x. N ) = 0 )
83	82	oveq2d 6017	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L ( M x. N ) ) = ( A /L 0 ) )
84		simpr 110	. . . . . 6 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> N = 0 )
85	84	oveq2d 6017	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L N ) = ( A /L 0 ) )
86	85	oveq2d 6017	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( A /L M ) x. ( A /L N ) ) = ( ( A /L M ) x. ( A /L 0 ) ) )
87	78, 83, 86	3eqtr4d 2272	. . 3 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
88	72, 87	jaodan 802	. 2 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
89		neanior 2487	. . 3 |- ( ( M =/= 0 /\ N =/= 0 ) <-> -. ( M = 0 \/ N = 0 ) )
90		lgsdi 15716	. . . 4 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
91	21, 90	syl3anl1 1319	. . 3 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
92	89, 91	sylan2br 288	. 2 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
93		zdceq 9522	. . . . 5 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> DECID M = 0 )
94	76, 23, 93	sylancl 413	. . . 4 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> DECID M = 0 )
95		zdceq 9522	. . . . 5 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
96	51, 23, 95	sylancl 413	. . . 4 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> DECID N = 0 )
97		dcor 941	. . . 4 |- (DECID M = 0 -> (DECID N = 0 -> DECID ( M = 0 \/ N = 0 ) ) )
98	94, 96, 97	sylc 62	. . 3 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
99		exmiddc 841	. . 3 |- (DECID ( M = 0 \/ N = 0 ) -> ( ( M = 0 \/ N = 0 ) \/ -. ( M = 0 \/ N = 0 ) ) )
100	98, 99	syl 14	. 2 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 \/ N = 0 ) \/ -. ( M = 0 \/ N = 0 ) ) )
101	88, 92, 100	mpjaodan 803	1 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   ifcif 3602   class class class wbr 4083  (class class class)co 6001   CCcc 7997   RRcr 7998   0cc0 7999   1c1 8000   x. cmul 8004   <_ cle 8182   2c2 9161   NN0cn0 9369   ZZcz 9446   ^cexp 10760   /Lclgs 15676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-frec 6537  df-1o 6562  df-2o 6563  df-oadd 6566  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-sup 7151  df-inf 7152  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-9 9176  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-fl 10490  df-mod 10545  df-seqfrec 10670  df-exp 10761  df-ihash 10998  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-clim 11790  df-proddc 12062  df-dvds 12299  df-gcd 12475  df-prm 12630  df-phi 12733  df-pc 12808  df-lgs 15677

This theorem is referenced by: (None)